Solve the equation. $\dfrac{dy}{dx}=4ye^x$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=e^{Ce^x}$ (Choice B) B $y=e^{e^x+C}$ (Choice C) C $y=e^{4e^x}+C$ (Choice D) D $y=Ce^{4e^x}$
Explanation: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=4ye^x \\\\ \dfrac{1}{y}\,dy&=4e^{x}\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} \dfrac{1}{y}\,dy&=4e^{x}\,dx \\\\ \int \dfrac{1}{y}\,dy&=\int 4e^{x}\,dx \\\\ \ln|y|&=4e^x+C_1 \\\\ e^{\ln|y|}&=e^{4e^x+C_1} \\\\ |y|&=e^{4e^x}e^{C_1} \\\\ y&=Ce^{4e^x} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=Ce^{4e^x}$